symplectic matrix

A real $2n\times 2n$ matrix $A\in {\mathrm{M}}_{2n}(ℝ)$ is a symplectic matrixif $AJ{A}^T=J$, where $A^T$ is the transpose of $A$, and$J\in \mathrm{O}(2n)$ is the orthogonal matrix

Here ${𝐈}_n\in {\mathrm{M}}_n(ℝ)$ is the identity $n\times n$ matrix and$\mathrm{𝟎}\in {\mathrm{M}}_n(ℝ)$ is the zero $n\times n$ matrix.

Symplectic matrices satisfy the followingproperties:

1. The determinant of a symplectic matrix equals one.

2. With standard matrix multiplication, symplectic $2n\times 2n$matrices form a group denoted by $\mathrm{Sp}(2n)$.

3. Suppose ,where $A,B,C,D$ are $n\times n$ matrices. Then$\mathrm{\Psi }$ is symplectic if and only if

   $$A{D}^T-B{C}^T=I,A{B}^T=B{A}^T,C{D}^T=D{C}^T.$$

4. If $X$ and $Y$ are real $n\times n$ matrices, then$U=X+iY$ is unitary if and only if is symplectic.